Paper Review

Aims

1. Use gaussian processes (GPs) to resample Type II(b) SNe light curves
2. Estimate some morphological parameters as defined by Pessi et al. (2019)
3. Assess the appropriateness of using GPs for this purpose
   • Goodness-of-fit of curves
   • Clustering of SNe by morphology parameters

Expecting a sharp divide between Type II and Type IIb SNe.

Type IIb SNe Light Curves

Type II SNe Light Curves

Gaussian Process

• mean function, \(\mu(t)\)
• covariance or kernel function, \(k(t_i, t_j)\)

\[\boldsymbol{Y} = \begin{bmatrix} Y_1 \\ \vdots \\ Y_n \end{bmatrix} \sim \mathcal{GP}(\boldsymbol{\mu}, \boldsymbol{\Sigma})\] where \(\boldsymbol{\mu} = \mu(t_i),\; \boldsymbol{\Sigma} = \mathrm{Cov}(Y_i, Y_j) = k(t_i, t_j)\quad i,j = 1, \dots, n\).

\[\textrm{Squared Exponential} \qquad k(\tau; \lambda) = A \exp\left\{-\frac{1}{2}\left( \frac{\tau}{\lambda}\right)^2\right\}\]

\[\textrm{Matern-3/2}\qquad k(\tau; \lambda) = \left(1 + \sqrt{3}\left(\frac{\tau}{\lambda}\right)\right) \exp\left\{-\sqrt{3}\left(\frac{\tau}{\lambda}\right) \right\}\]

Dataset

• Twenty-one “high quality” lightcurves from the Open Supernova Catalog accessible by API:
  1. evenly sampled
  2. well-studied explosion
  3. chosen by visual inspection (!)

Methodology

1. Fit a GP using different kernels (RBF, Matern-3/2)

2. Visually assess the goodness-of-fit

   • mean function (peak, plateau, linear decay)

3. Estimate the morphology parameters

   • \(t_\textrm{rise}\): time between explosion to maximum light
   • \(\Delta m_{40-30}\): mag. difference between phase 30 and 40
   • dm1: the earliest maximum of first derivative
   • dm2: the earliest minimum of second derivative

Results

Matern 3/2 vs RBF

Different GP implementations

First and Second Derivatives

NB: Matern-3/2 processes are only 1-time differentiable.

Clustering by \(t_\textrm{rise}\) and \(\Delta m_{40-30}\)

Conclusions

• Kernel choice is crucial, especially length-scale.
• Adding kernels together can partially fit complex behaviours at different scales.
• SN light curves are perhaps better fitted using non-stationary kernels that allow varying smoothness.
• Be cautious of different software implementations of kernel turning resulting in different results.
• Results are heavily dependent on the density of sampling.
• Still reproduced the clustering by Pessi et al. (2019)

Statistical claims needing caveats

• GPs tend to overfit
  • easy to say when the physics and behaviour are known.
  • selection of curves and goodness-of-fit was judged by visual inspection without consideration of variances.
• Don’t use models outside the range of their training data
  • Depends on context, e.g., what about forecasting?
• GP interpolations not suited to estimating dm1 and dm2
  • Matern-\(\nu\) kernels are only differentiable (“smooth”) up to \(\nu-1\) derivative.

Things to try

• Non-stationary GP kernels
• Uneven or sparsely sampled SNe light curves
• Incorporate uncertainty of observations